Problem Solving as State Space Search · Problem Solving as State Space Search • Formulate Goal...

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Brian Williams, Spring 03 1

Problem Solving as State Space Search

Brian C. Williams

16.410-13

Sep 15th, 2003Slides adapted from:6.034 Tomas Lozano Perez,Russell and Norvig AIMA


Complex missions must carefully:

• Plan complex sequences of actions

• Schedule tight resources

• Monitor and diagnose behavior

• Repair or reconfigure hardware.

Most AI problems, like these, may be formulated as state space search.

Courtesy of NASA/JPL-Caltech. http://www.jpl.nasa.gov.

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Assignments

• Remember: Problem set #1: Rules and Scheme due todaySeptember 15th, 2003.

• Reading: – Solving problems by searching: AIMA Ch. 3

• Homework for this lecture:– Problem set #2 due Monday, Sept. 22


Outline

• Problem Formulation– Problem solving as state space search

• Mathematical Model– Graphs and search trees

• Reasoning Algorithms– Depth and breadth-first search

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Early AI: What are the universal problem solving methods?

AstronautGooseGrainFox

Rover

Can the astronaut get its produce safely across the Martian canal?

• Astronaut + 1 item allowed in the rover.

• Goose alone eats Grain• Fox alone eats Goose

Simple Trivial


Problem Solving as State Space Search

• Formulate Goal– State

• Astronaut, Fox, Goose & Grain across river

• Formulate Problem– States

• Location of Astronaut, Fox, Goose & Grain at top or bottom river bank

– Operators• Move rover with astronaut & 1 or 0 items

to other bank.

• Generate Solution– Sequence of States

• Move(goose,astronaut), Move(astronaut), . . .

Image taken from NASA's website: http://www.nasa.gov.

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GrainFox

AstronautGoose

AstronautGrainFox

Goose

Goose

AstronautFox

Grain

AstronautGoose

GrainFox


Grain

AstronautGooseFox

AstronautGooseGrain

Fox

Fox

AstronautGooseGrain

AstronautGoose

Fox

Grain

GooseFox

AstronautGrain

GooseGrain

AstronautFox

AstronautGrain

GooseFox

AstronautFox

GooseGrain

GooseGrainFox

Astronaut

Astronaut

GooseGrainFox



GrainFox

AstronautGoose

AstronautGrainFox

Goose

Goose

AstronautFox

Grain

AstronautGoose

GrainFox


Grain

AstronautGooseFox

AstronautGooseGrain

Fox

Fox

AstronautGooseGrain

AstronautGoose

Fox

Grain

GooseFox

AstronautGrain

GooseGrain

AstronautFox

AstronautGrain

GooseFox

AstronautFox

GooseGrain

GooseGrainFox

Astronaut

Astronaut

GooseGrainFox

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Example: 8-Puzzle

• States:

• Operators:

• Goal Test:

5 4

6 1

7 3

8

2

1 2

8

3

7 6

4

5

Start Goal

integer location for each tile AND …

move empty square up, down, left, right

goal state as given


Planning as State Space Search:STRIPS Operator Representation

Effects specify how to change the set of assertions.

a anorth11

W0 W1

• Initial state:• (and (block a)

(block b)(block c) (on-table a) (on-table b) (clear a) (clear b) (clear c) (arm-empty))

• goal (partial state):• (and (on a b)

(on b c)))

• Available actions• Strips operators

precondition: (and (agent-at 1 1)(agent-facing north))

effect: (and (agent-at 1 2)(not (agent-at 1 1)))

North11

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STRIPS Action Schemata

(:operator pick-up:parameters ((block ?ob1)):precondition (and (clear ?ob1)

(on-table ?ob1) (arm-empty))

:effect (and (not (clear ?ob1))(not (on-table ?ob1))(not (arm-empty))(holding ?ob1)))

• Instead of defining: pickup-A and pickup-B and …

• Define a schema:Note: strips doesn’t

allowderived effects;

you must be complete!}


Outline




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Problem Formulation: A Graph

Operator

Link(edge)

State

Node(vertex)

DirectedGraph(one-way streets)

UndirectedGraph(two-way streets)


Examples of Graphs

San Fran

Boston

LA Dallas

Wash DC

Airline Routes

A B C

A B

C

A B

C

A

B

C

Put C on B

Put C on A

Put B on C

Put C on A

A

B

CPut A on C

Planning Actions

(graph of possible states of the world)

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A Graph

San Fran

Boston

LA Dallas

Wash DC

Airline Routes

A Graph G is represented as a pair <V,E>, where:

• V is a set of vertices

• E is a set of (directed) edges

An edge is a pair <v1, v2> of vertices, where

• v2 is the head of the edge,

•and v1 is the tail of the edge

< {Bos, SFO, LA, Dallas, Wash DC}

{<SFO, Bos>,

<SFO, LA>

<LA, Dallas>

<Dallas, Wash DC>

. . .} >


A Solution is a State Sequence:Problem Solving Searches Paths

C

S

B

GA

D

S

D

A

C

Represent searched paths using a tree.

<S, A, D, C>

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C

S

B

GA

D

S

D

A

C G




C

S

B

GA

D

S

D

BA

C G

C G

D

C G


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Search Trees

Root

Link(edge)

Node(vertex)


Search Trees

Parent(Ancestor)

Child(Descendant)

Siblings

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Outline





Classes of Search

Blind Depth-First Systematic exploration of whole tree

(uninformed) Breadth-First until the goal is found.

Iterative-Deepening

Heuristic Hill-Climbing Uses heuristic measure of goodness

(informed) Best-First of a node,e.g. estimated distance to.

Beam goal.

Optimal Branch&Bound Uses path “length” measure. Finds

(informed) A* “shortest” path. A* also uses heuristic

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Classes of Search

Blind Depth-First Systematic exploration of whole tree

(uninformed) Breadth-First until the goal is found.

Iterative-Deepening


Depth First Search (DFS)

S

D

BA

C G

C G

D

C G

Idea: •Explore descendants before siblings•Explore siblings left to right

1

2

3

4 5

6

7

8

9 10

11

S

A

D

C G

C

B

D

C G

G

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Breadth First Search (BFS)

S

D

BA

C G

C G

D

C G

Idea: •Explore relatives at same level before their children•Explore relatives left to right

1

2

4

8 9

5

3

6

10 11

7

S

A

D

C G

C

B

D

C G

G


Elements of Algorithm DesignDescription:

– stylized pseudo code, sufficient to analyze and implement the algorithm.

Analysis:• Soundness:

– is a solution returned by the algorithm guaranteed to be correct?

• Completeness: – is the algorithm guaranteed to find a solution when there is one?

• Time complexity: – how long does it take to find a solution?

• Space complexity: – how much memory does it need to perform search?

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Outline

• Problem Formulation: State space search

• Model: Graphs and search trees

• Reasoning Algorithms: DFS and BFS– A generic search algorithm

– Depth-first search example

– Handling cycles

– Breadth-first search example


Simple Search Algorithm

How do we maintain the Search State?• A set of partial paths explored thus far.

• An ordering on which partial path to expand next • called a queue Q.

How do we search? • Repeatedly:

• Select next partial path • Expand it.

• Terminate when goal found.

S

D

BA

C G

C G

D

C G

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Simple Search Algorithm• S denotes the start node• G denotes the goal node.• A partial path is a path from S to some node D,

• e.g., (D A S)

• The head of a partial path is the most recent node of the path, • e.g., D.

• The Q is a list of partial paths, • e.g. ((D A S) (C A S) …).

S

D

BA

C G

C G

D

C G


Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.

1. Initialize Q with partial path (S)2. If Q is empty, fail. Else, pick a partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else:

a) Remove N from Qb) Find all children of head(N) and

create all the one-step extensions of N to each child.c) Add all extended paths to Qd) Go to step 2.

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Outline





– Handling cycles



Depth First Search (DFS)

S

D

BA

C G

C G

D

C G

Idea: •Explore descendants before siblings•Explore siblings left to right

1

2

3

4 5

6

7

8

9 10

11

S

A

D

C G

C

B

D

C G

G

Where do we place the children on the queue?• Assume we pick first element of Q• Add path extensions to ? of Q

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Depth-First

Pick first element of Q; Add path extensions to front of Q

C

S

B

GA

D

Q

54321 (S)

1

18







C

S

B

GA

D

Q

54321 (S)

1

Depth-First


19


C

S

B

GA

D

Q

54321

(A S)(S)

1

Added paths in blue

Depth-First



C

S

B

GA

D

Q

54321

(A S) (B S)(S)

1

Added paths in blue

Depth-First


20







C

S

B

GA

D

Q

54321

(A S) (B S)(S)

1

Added paths in blue

2

Depth-First


21


C

S

B

GA

D

Q

54321

(C A S) (D A S) (B S)(A S) (B S)(S)

1

2

Depth-First


Added paths in blue


C

S

B

GA

D

Q

54321

(C A S) (D A S) (B S)(A S) (B S)(S)

1

2

Depth-First


Added paths in blue

22


C

S

B

GA

D

Q

54321

(C A S) (D A S) (B S)(A S) (B S)(S)

1

2

3

Depth-First


Added paths in blue


C

S

B

GA

D

Q

54321

(D A S) (B S)(C A S) (D A S) (B S)(A S) (B S)(S)

1

2

3

Depth-First


Added paths in blue

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C

S

B

GA

D

Q

54321


1

2

3

4

Depth-First


Added paths in blue


C

S

B

GA

D

Q

5

4321

(C D A S)(G D A S)(B S)


1

2

3

4

Depth-First


24







C

S

B

GA

D

Q

5

4321



1

2

3

4

Depth-First


25


C

S

B

GA

D

(G D A S)(B S)6

Q

5

4321



1

2

3

4

Depth-First



C

S

B

GA

D

(G D A S)(B S)6

Q

5

4321



1

2

3

4

Depth-First


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Outline





– Handling cycles



C

S

B

GA

D

Issue: Starting at S and moving top to bottom, will depth-first search ever reach G?

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C

S

B

GA

D

(G D A S)(B S)6

Q

5

4321



1

2

3

4

Depth-First

• C visited multiple times• Multiple paths to C, D & G

How much wasted effort can be incurred in the worst case?

Effort can be wasted in more mild cases


How Do We Avoid Repeat Visits?Idea:

• Keep track of nodes already visited.

• Do not place visited nodes on Q.

Does this maintain correctness?

• Any goal reachable from a node that was visited a second time would be reachable from that node the first time.

Does it always improve efficiency?

• Guarantees each node appears at most once at the head of a path in Q.

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1. Initialize Q with partial path (S) as only entry; set Visited = ( )2. If Q is empty, fail. Else, pick some partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else

a) Remove N from Qb) Find all children of head(N) not in Visited and

create all the one-step extensions of N to each child.c) Add to Q all the extended paths; d) Add children of head(N) to Visitede) Go to step 2.


Outline





– Handling cycles


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S

D

BA

C G

C G

D

C G

Idea: •Explore relatives at same level before their children•Explore relatives left to right

1

2

4

8 9

5

3

6

10 11

7

S

A

D

C G

C

B

D

C G

G

Where do we place the children on the queue?• Assume we pick first element of Q• Add path extensions to ? of Q


Breadth-FirstPick first element of Q; Add path extensions to end of Q

C

S

B

GA

D

6

VisitedQ

54321 S(S)

1

30



C

S

B

GA

D

6

VisitedQ

54321 S(S)

1



C

S

B

GA

D

6

VisitedQ

54321

A,B,S(A S) (B S)S(S)

1

31



C

S

B

GA

D

6

VisitedQ

54321

A,B,S(A S) (B S)S(S)

1

2



C

S

B

GA

D

6

VisitedQ

54321

C,D,B,A,S(B S) (C A S) (D A S)A,B,S(A S) (B S)S(S)

1

2

32



C

S

B

GA

D

6

VisitedQ

54321

C,D,B,A,S(B S) (C A S) (D A S)A,B,S(A S) (B S)S(S)

1

2

3



C

S

B

GA

D

G,C,D,B,A,S(G B S)6

VisitedQ

54321

G,C,D,B,A,S(D A S) (G B S)G,C,D,B,A,S(C A S) (D A S) (G B S)*C,D,B,A,S(B S) (C A S) (D A S)A,B,S(A S) (B S)S(S)

1

2

3

4

5

6

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Depth First Search (DFS)S

D

BA

C G

C G

D

C G

S

D

BA

C G

C G

D

C G


For each search type, where do we place the children on the queue?

Depth-first:

Add path extensions to front of Q

Pick first element of Q

Breadth-first:

Add path extensions to back of Q

Pick first element of Q


Summary• Most problem solving tasks may be

formulated as state space search.• Mathematical representations for search are

graphs and search trees.• Depth-first and breadth-first search may be

framed, among others, as instances of a generic search strategy.

• Cycle detection is required to achieve efficiency and completeness.

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Appendix


Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q

C

S

B

GA

D

(G B S) (C D A S) (G D A S)7(D B S) (G B S) (C D A S) (G D A S)6

Q

54321

(D A S) (D B S) (G B S)(C A S) (D A S) (D B S) (G B S)*(B S) (C A S) (D A S)(A S) (B S)(S)

1

2

3

4

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